(x',y') (x', y') (x, y) (x, y) Review: Rendering Pipeline. Review: Geometry Pipeline. Review: Graphics State. Transformations
|
|
|
- Anissa Hudson
- 6 years ago
- Views:
Transcription
1 Unverst of rtsh olum PS 34 omputer Grphcs Jn-pr 23 Tmr Munzner Trnsformtons Geometr Dtse Scn onverson Revew: Renderng Ppelne Model/Vew Trnsform. Teturng Lghtng Depth Test Perspectve Trnsform. lendng lppng Frmeuffer Revew: Grphcs Stte set the stte once remns untl overwrtten glolor3f... à set color to ellow glsetlerolor...2 à drk lue g glenlelight à turn on lght glenlegl_depth_test à hdden surf. Revew: Geometr Ppelne tell t how to nterpret geometr glegn<mode of geometrc prmtves> mode GL_TRINGLE GL_POLGON etc. feed t vertces glverte3f glverte3f.. -. glverte3f.. -. tell t ou re done glend Revew: GLUT: OpenGL Utlt Toolkt Redngs for Trnsformtons I-IV Trnsformtons smple portle wndow mnger openng wndows hndlng grphcs contets hndlng nput wth cllcks keord mouse wndow reshpe events tmng dle procesg dle events desgned for smll/medum sze pplctons FG hp 6 Trnsformton Mtrces ecept FG Sect 3.3 Scene Grphs 3rd ed: 2.2 R hp Vewng Vewng nd Modelng Trnsforms untl Vewng Trnsformtons Emples of ompog Severl Trnsformtons through uldng n rtculted Root rm R ppend Homogeneous oordntes nd Trnsformton Mtrces untl Perspectve Proecton R hp Dspl Lsts 2D Trnsformtons trnsformng n oect trnsformng ll ts ponts trnsformng polgon trnsformng ts vertces Mtr Representton Sclng Sclng Sclng represent 2D trnsformton wth mtr multpl mtr column vector ppl trnsformton to pont c d c d trnsformtons comned multplcton d e h c d f g k mtrces re effcent convenent w to represent sequence of trnsformtons 9 sclng coordnte mens multplng ech of ts components sclr unform sclng mens ths sclr s the sme for ll components: 2 non-unform sclng: dfferent sclrs per component: 2.5 how cn we represent ths n mtr form sclng operton: or n mtr form: sclng mtr 2 2D Rotton 2D Rotton From Trg Identtes 2D Rotton: nother Dervton 2D Rotton: nother Dervton counterclockwse RHS - φ r φ r φ r φ r φ Trg Identt r φ r φ r φ r φ Susttute
2 7 2D Rotton: nother Dervton 8 2D Rotton: nother Dervton 9 2D Rotton: nother Dervton 2 2D Rotton: nother Dervton 2 2D Rotton Mtr es to cpture n mtr form: even though q nd q re nonlner functons of q s lner comnton of nd s lner comnton of nd 22 Sher sher long s push ponts to rght n proporton to heght 23 Sher sher long s push ponts to rght n proporton to heght sh 24 Reflecton reflect cross s mrror 25 Reflecton reflect cross s mrror 26 2D Trnslton 27 2D Trnslton sclng mtr rotton mtr 28 2D Trnslton sclng mtr rotton mtr vector ddton mtr multplcton mtr multplcton 29 2D Trnslton sclng mtr rotton mtr d c trnslton multplcton mtr vector ddton mtr multplcton mtr multplcton 3 Lner Trnsformtons lner trnsformtons re comntons of sher scle rotte reflect propertes of lner trnsformtons stsfes Tst s T t T orgn mps to orgn lnes mp to lnes prllel lnes remn prllel rtos re preserved closed under composton d c d c 3 hllenge mtr multplcton for everthng ecept trnslton how to do everthng wth multplcton then ust do composton no specl cses homogeneous coordntes trck represent 2D coordntes wth 3-vector 32 Homogeneous oordntes our 2D trnsformton mtrces re now 33: Rotton Scle T T Trnslton use rghtmost column
3 Homogeneous oordntes Geometrcll Homogeneous oordntes Geometrcll homogeneous w w w w pont n 2D crte w w /w homogeneous w w w w w w pont n 2D crte weght w pont P n 3D homog. coords multples of w form lne L n 3D ll homogeneous ponts on L represent sme 2D crte pont emple: crte Homogeneous oordntes Summr dvde w to get /w /w proects lne to pont onto w plne lke normlzng one dmenson up ponts t nfnt these ponts cnnot e homogenzed les on - plne 3D Trnsformtons we ll see even more lter... P use 33 mtrces for 2D trnsformtons use 44 mtrces for 3D trnsformtons round s: z z n generl OpenGL commnd n rotte n z 3D Rotton n z c f w e 36 z z P orgn does not necessrl mp to orgn lnes mp to lnes prllel lnes remn prllel rtos re preserved closed under composton 3D Rotton out s m seem unntutve ut the mke grphcs opertons much eser llow ll ffne trnsformtons to e epressed through mtr multplcton d 35 s undefned w propertes of ffne trnsformtons when w consder t s drecton 34 lner trnsformtons trnsltons homogenze to convert homog. 3D pont to crte 2D pont: w w ffne trnsforms re comntons of w w /w 33 ffne Trnsformtons Homogeneous oordntes Geometrcll crte glrottefngle; glrottefngle; round s: z glrottefnglez; z z glrottefngle; 37 3D Sclng D Sher 3D Trnslton generl sher h sher h hz h hz hz hz hz h hz hz hz 4 Summr: Trnsformtons to vod mgut lws s sher long <s> n drecton of <s> z c z z c z < c > glsclefc; gltrnsltefc; 4 42 sherlongndrectonof h h sherlongndrectonof h h h sherlongndrectonof h sherlongndrectonof h h sherlongndrectonof h h sherlongndrectonof h h Rotte z z R z R T z R R z R z I Rotte z Rotte z T 2 T d d 2 d d 2 s d 2 S 2 S d 2 d 2 T 2 T d 2 d 2 so trnsltons dd s s 2 so scles multpl rotton 47 c z ompog Trnsformtons P T 2 P T 2 [T P ] [T 2 T] P where S s s sz S s s sz S s s sz S I s s sz sclng d d T T d d ompog Trnsformtons R s orthogonl 44 trnslton z 43 ompog Trnsformtons T z T z T z T z I sclec c z Undong Trnsformtons: Inverses trnsltec z R2 R so rottons dd 48
4 ompog Trnsformtons ompog Trnsformtons ompog Trnsformtons ompog Trnsformtons suppose we wnt suppose we wnt Rottez-9 suppose we wnt Rottez-9 Trnslte23 T T T T ut R R R R nd T R R T trnsltons commute rottons round sme s commute rottons round dfferent es do not commute p Rz9p p Rz9p p T23p rottons nd trnsltons do not commute ompog Trnsformtons suppose we wnt Rottez-9 Trnslte23 ompog Trnsformtons p TRp whch drecton to red ompog Trnsformtons p TRp whch drecton to red rght to left nterpret opertons wrt fed coordntes movng oect left to rght nterpret opertons wrt locl coordntes chngng coordnte sstem n OpenGL cnnot move oect once t s drwn oect specfed s set of coordntes wrt specfc coord ss ompog Trnsformtons p TRp whch drecton to red rght to left nterpret opertons wrt fed coordntes movng oect drw thng rotte thng -45 degrees wrt orgn trnslte t -2-3 over p Rz9p p T23p p T23Rz9p TRp ompog Trnsformtons p TRp whch drecton to red left to rght nterpret opertons wrt locl coordntes chngng coordnte sstem trnslte coordnte sstem 2 3 over rotte coordnte sstem 45 degrees wrt orgn drw oect n current coordnte sstem 57 ompog Trnsformtons p TRp whch drecton to red rght to left nterpret opertons wrt fed coordntes movng oect left to rght OpenGL ppelne orderng nterpret opertons wrt locl coordntes chngng coordnte sstem OpenGL updtes current mtr wth postmultpl gltrnsltef23; glrottef-9; glvertef; specf vector lst n fnl coordnte sstem frst mtr to ffect t s specfed second-to-lst 58 Interpretng Trnsformtons trnslte - 2 movng oect chngng coordnte sstem sme reltve poston etween oect nd ss vectors ntutve OpenGL 59 Mtr omposton mtrces re convenent effcent w to represent seres of trnsformtons generl purpose representton hrdwre mtr multpl mtr multplcton s ssoctve p T*R*S*p p T*R*S*p procedure correctl order our mtrces multpl mtrces together result s one mtr multpl vertces ths mtr ll vertces esl trnsformed wth one mtr multpl 6 Rotton out Pont: Movng Oect Rotton: hngng oordnte Sstems Rotton: hngng oordnte Sstems Rotton: hngng oordnte Sstems rotte out p : p trnslte p to orgn rotte out orgn trnslte p ck sme emple: rotton round rtrr center rotton round rtrr center step : trnslte coordnte sstem to rotton center rotton round rtrr center step 2: perform rotton T zrztz T zrztz T zrztz T zrztz
5 Rotton: hngng oordnte Sstems rotton round rtrr center Generl Trnsform omposton Rotton out n rtrr s trnsformton of geometr nto coordnte sstem where operton ecomes smpler step 3: ck to orgnl coordnte sstem tpcll trnslte to orgn perform operton rtrr Rotton s defned two ponts trnslte pont to the orgn rotte to lgn s wth z-s or or perform rotton undo lgnng rottons undo trnslton rtrr rotton: chnge of ss gven two orthonorml coordnte sstems nd s locton n the coordnte sstem s z... trnsform geometr ck to orgnl coordnte sstem T zr z T z rtrr Rotton z rtrr Rotton z rtrr rotton: chnge of ss z c c cz z rtrr rotton: chnge of ss gven two orthonorml coordnte sstems nd Trnsformton Herrches c c cz stores mtr t ech level wth ncrementl trnsform from prent s coordnte sstem Trnsformton Herrches gven two orthonorml coordnte sstems nd s locton n the coordnte sstem s z... scene m hve herrch of coordnte sstems s locton n the coordnte sstem s z... trnsformton from one to the other s mtr R whose columns re : R z c c z cz scene grph z rod strpe strpe2... cr cr2... w w2 w3 w Trnsformton Herrch Emple Trnsformton Herrch Emple 2 Trnsformton Herrches Demo drw sme 3D dt wth dfferent trnsformtons: nstncng Mtr Stcks trnsforms ppl to grph nodes eneth chllenge of vodng unnecessr computton world ug nverse to return to orgn computng ncrementl T -> T2 torso LUleg RUleg LUrm RUrm LLleg RLleg LLrm RLrm Lfoot Rfoot Lhnd Rhnd hed Mtr Stcks glpushmtr glscle3f222 gltrnslte3f DrwSqure glpopmtr 76 Trnsformton Herrch Emple 3 ccumulton of numercl errors prctcl ssues n grphcs hrdwre depth of mtr stcks s lmted tpcll 6 for model/vew nd out 4 for proectve mtr Fh glpopmtr; FW } 77 World coordntes no need to compute nverse mtrces ll the tme modulrze chnges to ppelne stte vods ncrementl chnges to coordnte sstems glsclefkkk; glegngl_line_loop; glverte3f; glverte3f; glverte3f; glverte3f; glend; DrwSqure T3 dvntges vod drwlockflot k { glpushmtr; glpopmtr 75 Mtr Stcks push/pop ensures no coord sstem chnge D scle222 trns D scenegrphs.html Modulrzton drwng scled squre T2 T trns.3 rotz glpushmtr Oect coordntes F Fh gllodidentt; gltrnsltef4; glpushmtr; glrottef45; gltrnsltef2; glsclef2; gltrnslte; glpopmtr; 8
6 Trnsformton Herrch Emple 4 Herrchcl Modellng gltrnslte3f; glrottef ; Drwod; glpushmtr; gltrnslte3f7; DrwHed; glpopmtr; glpushmtr; gltrnslte2.55.5; glrottef 2; DrwUrm; gltrnslte-3.5; glrottef 3; DrwLrm; glpopmtr;... drw other rm Dspl Lsts precomple/cche lock of OpenGL code for reuse dvntges defne oect once nstntte multple copes trnsformton prmeters often good control knos mntn structurl constrnts f well-desgned lmttons epressvt: not lws the est controls cn t do closed knemtc chns usull more effcent thn mmedte mode ect optmztons depend on drver good for multple nstnces of sme oect // Drw od gltrnsltef.f.75f.f; glutsoldsphere.75f22; // Drw Hed gltrnsltef.f.f.f; glutsoldsphere.25f22; collson detecton cn e nested herrchcll self-ntersecton wlk through wlls snowmn emple 82 Instntte Mn Snowmen fornt -3; < 3; { glpushmtr; gltrnsltef*. *.; // ll the functon to drw snowmn drwsnowmn; glpopmtr; 36K polgons 55 FPS 85 Trnsformng Geometrc Oects omputng Normls lnes polgons mde up of vertces 86 snowmndl cretedl; fornt -3; < 3; fornt -3; < 3; { glpushmtr; gltrnsltef*. *.; gllllstsnowmndl; 36K glpopmtr; } z drecton specfng orentton of polgon w mens drecton wth homogeneous coords vs. w for ponts/vectors of oect vertces normls re trcker must e normlzed to unt length cn compute f not suppled wth oect N P 89 Trnsformng Normls new plne 2 norml s drecton of lne -2 or 2 not perpendculr to plne should e drecton of 2-93 m3 m22 m23 m32 m33 Trnsformng Normls T T Tz z nonunform sclng does not work - plne lne norml: [-] drecton of lne - gnore normlzton for now 9 Fndng orrect Norml Trnsform cz d P P MP N N QN N T P QN T MP N T QT MP QT M I T Q M Redng for Net Topc: Vewng trnsform plne N P wth dot product N T P mtr multpl requres trnspose N P c z d w eplct form: plne m2 9 plne s ll ponts perpendculr to norml trnsformed norml: [2-] m m2 m3 88 these ll mntn drecton N P2 P P3 P Plnes nd Normls ppl nonunform scle: stretch long 2 87 trnsltons OK: w mens unffected rottons OK unform sclng OK P3 P2 polgons 53 FPS so f ponts trnsformed mtr M cn we ust trnsform norml vector M too used for lghtng does ths work for everthng no Trnsformng Normls Trnsformng Normls N norml trnsform the vertces nterpolte etween 84 Mkng Dspl Lsts fornt -3; < 3; // Drw Nose glolor3f.f.5f.5f; glrottef.f.f.f.f; glutsoldone.8f.5f2; } 83 GLunt cretedl { GLunt snowmndl; // rete the d for the lst snowmndl glgenlsts; glnewlstsnowmndlgl_ompile; drwsnowmn; glendlst; returnsnowmndl; } // Drw 36 Snowmen // Drw Ees glpushmtr; glolor3f.f.f.f; gltrnsltef.5f.f.8f; glutsoldsphere.5f; gltrnsltef-.f.f.f; glutsoldsphere.5f; glpopmtr; } 2 2 dspl lsts persst cross multple frmes nterctve grphcs: oects redrwn ever frme from new vewpont from movng cmer keep hnd on hp One Snowmn glolor3f.f.f.f; good for sttc oects redrwn often cn t do other constrnts 8 vod drwsnowmn { ut cnnot chnge contents not prmetrzle Dspl Lsts FG hpter 7 Vewng FG Secton 6.3. Wndowng Trnsforms gven M wht should Q e st perpendculr susttute from ove R rest of hp Vewng R rest of pp Homogeneous oords T T T NT P f Q T M I thus the norml to n surfce cn e trnsformed the nverse trnspose of the modellng trnsformton 95 96
Multiple view geometry
EECS 442 Computer vson Multple vew geometry Perspectve Structure from Moton - Perspectve structure from moton prolem - mgutes - lgerc methods - Fctorzton methods - Bundle djustment - Self-clrton Redng:
Announcements. Image Formation: Outline. The course. Image Formation and Cameras (cont.)
nnouncements Imge Formton nd Cmers (cont.) ssgnment : Cmer & Lenses, gd Trnsformtons, nd Homogrph wll be posted lter tod. CSE 5 Lecture 5 CS5, Fll CS5, Fll CS5, Fll The course rt : The phscs of mgng rt
Review of linear algebra. Nuno Vasconcelos UCSD
Revew of lner lgebr Nuno Vsconcelos UCSD Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8)
Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
Principle Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
Definition of Tracking
Trckng Defnton of Trckng Trckng: Generte some conclusons bout the moton of the scene, objects, or the cmer, gven sequence of mges. Knowng ths moton, predct where thngs re gong to project n the net mge,
ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
Physics for Scientists and Engineers I
Phscs for Scentsts nd Engneers I PHY 48, Secton 4 Dr. Betr Roldán Cuen Unverst of Centrl Flord, Phscs Deprtment, Orlndo, FL Chpter - Introducton I. Generl II. Interntonl Sstem of Unts III. Converson of
Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ]
![Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ]](/thumbs/92/109192837.jpg "Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ]")
Uses of trnsformtions 3D trnsformtions Modeling: position nd resize prts of complex model; Viewing: define nd position the virtul cmer Animtion: define how objects move/chnge with time y y Sclr (dot) product
4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
7.2 Volume. A cross section is the shape we get when cutting straight through an object.
7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A
VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors
1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
Pyramid Algorithms for Barycentric Rational Interpolation
Pyrmd Algorthms for Brycentrc Rtonl Interpolton K Hormnn Scott Schefer Astrct We present new perspectve on the Floter Hormnn nterpolnt. Ths nterpolnt s rtonl of degree (n, d), reproduces polynomls of degree
VECTORS AND TENSORS IV.1.1. INTRODUCTION
Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 05 IV. VECTORS AND TENSORS IV... INTRODUCTION In mthemtcs nd mechncs, we he to operte wth qunttes whch requre dfferent mthemtcl ojects
DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
CISE 301: Numerical Methods Lecture 5, Topic 4 Least Squares, Curve Fitting
CISE 3: umercl Methods Lecture 5 Topc 4 Lest Squres Curve Fttng Dr. Amr Khouh Term Red Chpter 7 of the tetoo c Khouh CISE3_Topc4_Lest Squre Motvton Gven set of epermentl dt 3 5. 5.9 6.3 The reltonshp etween
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
Things to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
SVMs for regression Multilayer neural networks
Lecture SVMs for regresson Muter neur netors Mos Husrecht mos@cs.ptt.edu 539 Sennott Squre Support vector mchne SVM SVM mmze the mrgn round the seprtng hperpne. he decson functon s fu specfed suset of
Lecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
PART 1: VECTOR & TENSOR ANALYSIS
PART : VECTOR & TENSOR ANALYSIS wth LINEAR ALGEBRA Obectves Introduce the concepts, theores, nd opertonl mplementton of vectors, nd more generlly tensors, n dvnced engneerng nlyss. The emphss s on geometrc
Lecture 3 Camera Models 2 & Camera Calibration. Professor Silvio Savarese Computational Vision and Geometry Lab
Lecture Cer Models Cer Clbrton rofessor Slvo Svrese Coputtonl Vson nd Geoetry Lb Slvo Svrese Lecture - Jn 7 th, 8 Lecture Cer Models Cer Clbrton Recp of cer odels Cer clbrton proble Cer clbrton wth rdl
ME 501A Seminar in Engineering Analysis Page 1
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 More oundr-vlue Prolems nd genvlue Prolems n Os Lrr retto Menl ngneerng 5 Semnr n ngneerng nlss ovemer 9, 7 Outlne Revew oundr-vlue prolems Soot
Demand. Demand and Comparative Statics. Graphically. Marshallian Demand. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert
Demnd Demnd nd Comrtve Sttcs ECON 370: Mcroeconomc Theory Summer 004 Rce Unversty Stnley Glbert Usng the tools we hve develoed u to ths ont, we cn now determne demnd for n ndvdul consumer We seek demnd
LINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
Proof that if Voting is Perfect in One Dimension, then the First. Eigenvector Extracted from the Double-Centered Transformed
Proof tht f Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted from the Doule-Centered Trnsformed Agreement Score Mtrx hs the Sme Rn Orderng s the True Dt Keth T Poole Unversty of Houston
Partially Observable Systems. 1 Partially Observable Markov Decision Process (POMDP) Formalism
CS294-40 Lernng for Rootcs nd Control Lecture 10-9/30/2008 Lecturer: Peter Aeel Prtlly Oservle Systems Scre: Dvd Nchum Lecture outlne POMDP formlsm Pont-sed vlue terton Glol methods: polytree, enumerton,
Announcements. Texture Mapping PA4. Geometric Registration
Ross Beerdge Bre Drper Annonements Tetre Mppng Letre 5 Tesd, Noemer 7 th 5 PA grdes eng djsted Penltes for flppng dmensons to e remoed PA4 hnded ot tod De Tesd, De. 8 th See net slde for detls /7/5 PA4
The Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
The Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
Physics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
Introduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
Engineering Tensors. Friday November 16, h30 -Muddy Charles. A BEH430 review session by Thomas Gervais.
ngneerng Tensors References: BH4 reew sesson b Thoms Gers tgers@mt.ed Long, RR, Mechncs of Solds nd lds, Prentce-Hll, 96, pp - Deen, WD, nlss of trnsport phenomen, Oford, 998, p. 55-56 Goodbod, M, Crtesn
An Introduction to Support Vector Machines
An Introducton to Support Vector Mchnes Wht s good Decson Boundry? Consder two-clss, lnerly seprble clssfcton problem Clss How to fnd the lne (or hyperplne n n-dmensons, n>)? Any de? Clss Per Lug Mrtell
5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I
5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can
Rigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement.
Phscs 01, Lecture 4 Toda s Topcs n Vectors chap 3) n Scalars and Vectors n Vector ddton ule n Vector n a Coordnator Sstem n Decomposton of a Vector n Epected from prevew: n Scalars and Vectors, Vector
Chapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
Computer Graphics (CS 4731) Lecture 7: Linear Algebra for Graphics (Points, Scalars, Vectors)
Computer Grphics (CS 4731) Lecture 7: Liner Alger for Grphics (Points, Sclrs, Vectors) Prof Emmnuel Agu Computer Science Dept. Worcester Poltechnic Institute (WPI) Annoncements Project 1 due net Tuesd,
Least squares. Václav Hlaváč. Czech Technical University in Prague
Lest squres Václv Hlváč Czech echncl Unversty n Prgue hlvc@fel.cvut.cz http://cmp.felk.cvut.cz/~hlvc Courtesy: Fred Pghn nd J.P. Lews, SIGGRAPH 2007 Course; Outlne 2 Lner regresson Geometry of lest-squres
6 Roots of Equations: Open Methods
HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 6 Roots of Equtons: Open Methods Smple Fed-Pont Iterton Newton-Rphson Secnt Methods MATLAB Functon: fzero Polynomls Cse Study: Ppe Frcton Brcketng
A general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex.
Lnr lgr Vctors gnrl -dmnsonl ctor conssts of lus h cn rrngd s column or row nd cn rl or compl Rcll -dmnsonl ctor cn rprsnt poston, loct, or cclrton Lt & k,, unt ctors long,, & rspctl nd lt k h th componnts
DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
FUNDAMENTALS ON ALGEBRA MATRICES AND DETERMINANTS
Dol Bgyoko (0 FUNDAMENTALS ON ALGEBRA MATRICES AND DETERMINANTS Introducton Expressons of the form P(x o + x + x + + n x n re clled polynomls The coeffcents o,, n re ndependent of x nd the exponents 0,,,
2.12 Pull Back, Push Forward and Lie Time Derivatives
Secton 2.2 2.2 Pull Bck Push Forwrd nd e me Dertes hs secton s n the mn concerned wth the follown ssue: n oserer ttched to fxed sy Crtesn coordnte system wll see mterl moe nd deform oer tme nd wll osere
Torsion, Thermal Effects and Indeterminacy
ENDS Note Set 7 F007bn orson, herml Effects nd Indetermncy Deformton n orsonlly Loded Members Ax-symmetrc cross sectons subjected to xl moment or torque wll remn plne nd undstorted. At secton, nternl torque
Linear and Nonlinear Optimization
Lner nd Nonlner Optmzton Ynyu Ye Deprtment of Mngement Scence nd Engneerng Stnford Unversty Stnford, CA 9430, U.S.A. http://www.stnford.edu/~yyye http://www.stnford.edu/clss/msnde/ Ynyu Ye, Stnford, MS&E
x=0 x=0 Positive Negative Positions Positions x=0 Positive Negative Positions Positions
Knemtc Quntte Lner Moton Phyc 101 Eyre Tme Intnt t Fundmentl Tme Interl Defned Poton x Fundmentl Dplcement Defned Aerge Velocty g Defned Aerge Accelerton g Defned Knemtc Quntte Scler: Mgntude Tme Intnt,
Quiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
Module 3: Element Properties Lecture 5: Solid Elements
Modue : Eement Propertes eture 5: Sod Eements There re two s fmes of three-dmenson eements smr to two-dmenson se. Etenson of trngur eements w produe tetrhedrons n three dmensons. Smr retngur preeppeds
LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN FRACTAL HEAT TRANSFER
Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S3 LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN
So far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
Review: Transformations. Transformations - Viewing. Transformations - Modeling. world CAMERA OBJECT WORLD CSE 681 CSE 681 CSE 681 CSE 681
Revew: Trnsforons Trnsforons Modelng rnsforons buld cople odels b posonng (rnsforng sple coponens relve o ech oher ewng rnsforons plcng vrul cer n he world rnsforon fro world coordnes o cer coordnes Perspecve
Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
Kinematics Quantities. Linear Motion. Coordinate System. Kinematics Quantities. Velocity. Position. Don t Forget Units!
Knemtc Quntte Lner Phyc 11 Eyre Tme Intnt t Fundmentl Tme Interl t Dened Poton Fundmentl Dplcement Dened Aerge g Dened Aerge Accelerton g Dened Knemtc Quntte Scler: Mgntude Tme Intnt, Tme Interl nd Speed
Support vector machines for regression
S 75 Mchne ernng ecture 5 Support vector mchnes for regresson Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre S 75 Mchne ernng he decson oundr: ˆ he decson: Support vector mchnes ˆ α SV ˆ sgn αˆ SV!!: Decson
Effects of polarization on the reflected wave
Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton
Variable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
Mathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
Geometric Correction or Georeferencing
Geoetrc Correcton or Georeferencng GEOREFERENCING: fro ge to p Coordntes on erth: (λ, φ) ge: (, ) p: (, ) rel nteger Trnsfortons (nvolvng deforton): erth-to-ge: χ erth-to-p: ψ (crtogrphc proecton) ge-to-p:
Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no
hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton
Three hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Wednesday 16th January 2013 Time: 09:45-12:45
Three hours UNIVERSITY OF MANHESTER SHOOL OF OMPUTER SIENE Prllel Progrms nd ther Performnce Dte: Wednesdy 16th Jnury 2013 Tme: 09:45-12:45 Plese nswer ny TWO Questons from the FOUR Questons provded. Ths
Lecture 8: Camera Calibration
Lecture 8: Cer Clbrton rofessor Fe-Fe L Stnford Vson Lb Fe-Fe L 9-Oct- Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbrton Vnshng ponts nd lnes (roble Set (Q)) Redng: [F]
Section 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
Unit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
GAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
To Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y
Foundtions of Computer Grphics (Spring 2010) CS 184, Lecture 2: Review of Bsic Mth http://inst.eecs.erkeley.edu/~cs184 o Do Complete Assignment 0 Downlod nd compile skeleton for ssignment 1 Red instructions
Transform Coding. C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University
Trnsform Codng C.M. Lu Perceptul Sgnl Processng Lb College of Computer Scence Ntonl Cho-Tung Unversty http://www.cse.nctu.edu.tw/~cmlu/courses/compresson/ Offce: EC538 (03)573877 cmlu@cs.nctu.edu.tw Motvtng
Dynamic Power Management in a Mobile Multimedia System with Guaranteed Quality-of-Service
Dynmc Power Mngement n Moble Multmed System wth Gurnteed Qulty-of-Servce Qnru Qu, Qng Wu, nd Mssoud Pedrm Dept. of Electrcl Engneerng-Systems Unversty of Southern Clforn Los Angeles CA 90089 Outlne! Introducton
2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
APPLICATIONS OF DEFINITE INTEGRALS
Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We
5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
CALIBRATION OF SMALL AREA ESTIMATES IN BUSINESS SURVEYS
CALIBRATION OF SMALL AREA ESTIMATES IN BUSINESS SURVES Rodolphe Prm, Ntle Shlomo Southmpton Sttstcl Scences Reserch Insttute Unverst of Southmpton Unted Kngdom SAE, August 20 The BLUE-ETS Project s fnnced
SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
INTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
Introduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
8. INVERSE Z-TRANSFORM
8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere
MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
Substitution Matrices and Alignment Statistics. Substitution Matrices
Susttuton Mtrces nd Algnment Sttstcs BMI/CS 776 www.ostt.wsc.edu/~crven/776.html Mrk Crven crven@ostt.wsc.edu Ferur 2002 Susttuton Mtrces two oulr sets of mtrces for roten seuences PAM mtrces [Dhoff et
Error Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:
More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ + + + mn mnmzes χ, nd
Linear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
CIS587 - Artificial Intelligence. Uncertainty CIS587 - AI. KB for medical diagnosis. Example.
CIS587 - rtfcl Intellgence Uncertnty K for medcl dgnoss. Exmple. We wnt to uld K system for the dgnoss of pneumon. rolem descrpton: Dsese: pneumon tent symptoms fndngs, l tests: Fever, Cough, leness, WC
SVMs for regression Non-parametric/instance based classification method
S 75 Mchne ernng ecture Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre SVMs for regresson Non-prmetrc/nstnce sed cssfcton method S 75 Mchne ernng Soft-mrgn SVM Aos some fet on crossng the seprtng hperpne
Lecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
Work and Energy (Work Done by a Varying Force)
Lecture 1 Chpter 7 Physcs I 3.5.14 ork nd Energy (ork Done y Vryng Force) Course weste: http://fculty.uml.edu/andry_dnylov/techng/physcsi Lecture Cpture: http://echo36.uml.edu/dnylov13/physcs1fll.html
Geometrical Transformations
Geometricl Trnsformtions Did Atkinson D.Atkinson@cl.c.k Compttionl Aspects of MRI References Fole, n Dm, Feiner, Hghes. Compter Grphics: Principles nd Prctice. Chpter 5. Wolfrm MthWorld http://mthworld.wolfrm.com/
AP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
DA 3: The Mean Value Theorem
Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit
? plate in A G in
Proble (0 ponts): The plstc block shon s bonded to rgd support nd to vertcl plte to hch 0 kp lod P s ppled. Knong tht for the plstc used G = 50 ks, deterne the deflecton of the plte. Gven: G 50 ks, P 0
Course Review Introduction to Computer Methods
Course Revew Wht you hopefully hve lerned:. How to nvgte nsde MIT computer system: Athen, UNIX, emcs etc. (GCR). Generl des bout progrmmng (GCR): formultng the problem, codng n Englsh trnslton nto computer
Section 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
Vectors and Tensors. R. Shankar Subramanian. R. Aris, Vectors, Tensors, and the Equations of Fluid Mechanics, Prentice Hall (1962).
005 Vectors nd Tensors R. Shnkr Subrmnn Good Sources R. rs, Vectors, Tensors, nd the Equtons of Flud Mechncs, Prentce Hll (96). nd ppendces n () R. B. Brd, W. E. Stewrt, nd E. N. Lghtfoot, Trnsport Phenomen,
PHYS 2421 Fields and Waves
PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4